Introduction to Abstract Algebra, Set, 4th Edition

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Praise for the Third Edition

". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."—Zentralblatt MATH

The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text.

The Fourth Edition features important concepts as well as specialized topics, including:

  • The treatment of nilpotent groups, including the Frattini and Fitting subgroups

  • Symmetric polynomials

  • The proof of the fundamental theorem of algebra using symmetric polynomials

  • The proof of Wedderburn's theorem on finite division rings

  • The proof of the Wedderburn-Artin theorem

Throughout the book, worked examples and real-world problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book's exercises.

Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics.

Preface ix

Acknowledgment xv

Notations Used in the Text xvii

A Sketch of the History of Algebra to 1929 xxi

Preliminaries 1

Proofs 1

Sets 5

Mappings 9

Equivalences 17

Integers and Permutations 22

Induction 22

Divisors and Prime Factorization 30

Integers Modulo n 41

Permutations 51

An Application to Cryptography 63

Groups 66

Binary Operations 66

Groups 73

Subgroups 82

Cyclic Groups and the Order of an Element 87

Homomorphisms and Isomorphisms 95

Cosets and Lagrange's Theorem 105

Groups of Motions and Symmetries 114

Normal Subgroups 119

Factor Groups 127

The Isomorphism Theorem 133

An Application to Binary Linear Codes 140

Rings 155

Examples and Basic Properties 155

Integral Domains and Fields 166

Ideals and Factor Rings 174

Homomorphisms 183

Ordered Integral Domains 193

Polynomials 196

Polynomials 196

Factorization of Polynomials over a Field 209

Factor Rings of Polynomials over a Field 222

Partial Fractions 231

Symmetric Polynomials 233

Formal Construction of Polynomials 243

Factorization in Integral Domains 246

Irreducibles and Unique Factorization 247

Principal Ideal Domains 259

Fields 268

Vector Spaces 269

Algebraic Extensions 277

Splitting Fields 285

Finite Fields 293

Geometric Constructions 299

The Fundamental Theorem of Algebra 304

An Application to Cyclic and BCH Codes 305

Modules over Principal Ideal Domains 318

Modules 318

Modules over a PID 327

p-Groups and the Sylow Theorems 341

Factors and Products 341

Cauchy's Theorem 349

Group Actions 356

The Sylow Theorems 364

Semidirect Products  371

An Application to Combinatorics 375

Series of Subgroups 381

The Jordan-Holder Theorem 382

Solvable Groups 387

Nilpotent Groups 394

Galois Theory 401

Galois Groups and Separability 402

The Main Theorem of Galois Theory 410

Insolvability of Polynomials 423

Cyclotomic Polynomials and Wedderburn's Theorem 430

Finiteness Conditions for Rings and Modules 435

Wedderburn's Theorem 435

The Wedderburn-Artin Theorem 444

Appendices

Complex Numbers 455

Matrix Arithmetic 462

Zorn's Lemma 467

Proof of the Recursion Theorem 471

Bibliography 473

Selected Answers 475

Index 499

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